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From the studies of the history of economic thought, we can learn that many theorems of the modern general equilibrium theory were well anticipated in the classical economics and the economics of the marginal revolution. For example, first, what Johann Heinrich von Thunen (1783–1850) did, in his strange theory of natural wage, can be interpreted as an early, pioneering attempt to use the so-called Negishi method (1960), which is now intensively used for the proof of the existence theorem and the numerical calculation of a general equilibrium. Second, as is well known, no exchanged transactions are permitted out of equilibria in the famous Walrasian tatonnement adjustment process towards market equilibria. Against this, the importance of transactions carried out at disequilibria is emphasized in the studies of the so-called Hahn–Negishi non-tatonnement process (1962). As a matter of fact, in the classical economics, this is exactly what William Thomas Thornton (1813–1380) insisted against the authority of John Stuart Mill (1800–1873). Finally, subjectively perceived (often kinked) demand curves are considered in my proof of the existence of a general equilibrium in the case of monopolistic competition (1961). To my surprise, however, I found later that such demand curves were already hidden in Adam Smith's (1723–1790) consideration of markets and the division of labor, and that the increase of demand never fails to lower the price of goods.

No one can deny the important contributions made by Johann Heinrich von Thunen (1783-1850), a pioneer of the marginal revolution in economics, as an independent discoverer of the marginal productivity theory and as the father of the economics of space. Commemorating the two hundredth anniversary of his birth, Samuelson (1983) analyzed these contributions beautifully and skillfully. Von Thunen is, however, also known by the formula of natural wage inscribed on his grave stone. Classical economists before the marginal revolution explained the wage as the subsistence wage, while modern economists after the revolution emphasized the importance of the labor productivity in their theory of wage. In between, Thunen demonstrated the natural wage as the geometric mean of the subsistence wage and the average product of a laborer. Samuelson, who evaluated Thunen's positivistic economics (the marginal productivity theory and the economics of space) very high, interpreted the natural wage of Thunen as the socially optimal wage from the point of view of the welfare economics, and criticized that it is a major felony and a crime against normative economics.In Negishi (1990), we defended Thunen against Samuelson. Following an interpretation suggested by Yamada (1934), a pioneering study of Thunen in Japan, the natural wage is derived as an equilibrium wage in a competitive stationary economy, which Thunen himself studied in his famous Der isolierte Staat. In other words, we defended the theory of natural wage as a positive theory against Samuelson's criticism which regarded it as a normative theory. To be more decisive, however, as a rejoinder to Samuelson's critical comment that Thunen maximized a strange, irrational social welfare function, we have to argue further that the so-called Thunen's social welfare function itself is a rational, reasonable function, given his own model of a competitive stationary economy. In other words, what Thunen did can be interpreted as an early, pioneering attempt to use the so-called Negishi method, Negishi approach, or Negishi theorem, which is now intensively used for the proof of the existence of a general equilibrium and the numerical calculation of a general equilibrium. See Negishi (1960), Kehoe (1991), Mas-Colell and Zame (1991) and Ginsburgh and Keyzer (1997).

In Negishi (1986), I developed my view on the so-called Mill's recantation of the wages fund doctrine, the short-run theory of wages in the classical school of the economics. According to a survey article on the recent John Stuart Mill interpretations, “a further twist to the debate has been given by Negishi (1986). He reminds us that Mill, in the 1871 edition of the Principles, retracted some of his earlier enthusiasm for Thornton. There he in effect rejected the idea that disequilibrium trades in any way undermine the importance of the principle of the equality of supply and demand, and cautioned that it was too early to adduce any firm conclusions from the discussion engendered by Thornton's book”(De Marchi, 1988).I have three purposes in this present article. The first one is to offer a Japanese version of Negishi (1986), which I think it worthwhile, since there are still no Japanese translation available of the related literature referred and quoted there, i. e., Thornton (1869), Mill (1869) and Thornton (1870). Secondly, I wish to record the appearance of some related literature published in 1990's, like Ekelund (1997), Forget (1991), Fukagai (1995), Mawatari (1997), Mirowsky (1990), Negishi (1998), Vint (1994), and White (1994). Particularly, I wish to make some detailed rejoinders to comments given by Mawatari (1997).The final one is to consider, from the point of view of the modern economic theory (Arrow and Hahn (1971), pp. 324-346, and Negishi (1972), pp. 207-227), how Mill should have replied to Thornton, who asked the significance of a small amount of trade at equilibrium prices, which are arrived after a great bulk of goods were already sold at disequilibrium prices (Thornton, 1869, pp. 53-54; 1870, p. 65). As is easily seen at a glance at Figures 6, 7 and 8 in the text of the present article, i. e., those of box diagram and of utility frontier, the importance of the final small amount of trade at equilibrium prices is that the Pareto optimality is assured.

The nature and significance of input–output analysis or interindustry analysis, which was initiated by W. Leontief, can be interpreted in many different ways. It can be seen as the generalization of Keynesian multiplier theory, or as the modern version of Quesnay’s Tableau economique or Marx’s reproduction scheme, or as a degenerated linear programming model. From our point of view, however, it can be considered as an empirically oriented simplified version of the general equilibrium theory. While the importance of the analysis of interindustry input–output relations is duly emphasized, consumer demands are assumed to be given by outside information. The optimization and substitutability in production are assumed away by the use of the no-joint-output constant-coefficient production functions which are, however, numerically estimated by the data available from input–output tables. The input–output analysis can, therefore, provide numerical answers for a number of interesting problems of the general equilibrium of an economy.

The basic economic model of the so-called Heckscher–Ohlin Theory is the two- country two-commodity two-factor model where production functions are linear-homogeneous and the endowment of two factors of production is exogenously given for each country. Besides the factor price equalization theorem and Heckscher–Ohlin theorem discussed in Chap. 10, we can have also two additional well-known theorems derived from this Heckscher–Ohlin model. The first one is the so-called Stolper–Samuelson Theorem (Stolper and Samuelson (1941)) which considers the effects of an import tariff on the factor prices. In other words, it concerns with the changes in factor prices caused by a change in the relative price of two commodities. The second one is called Rybczynski Theorem (Rybczynski 1955) which deals with the changes in the scale of production of two industries caused by a change in the factor endowment in the case of a small country. This is the problem of the effects of the economic growth on the relative scale of the two domestic industries, when the relative price of two commodities is assumed to be constant.

In the current theory of general economic equilibrium, recontracting and tâtonnement (a French word meaning ‘groping’) are used interchangeably to denote a simplifying assumption that no actual transactions, and therefore no production and consumption activities, take place at disequilibria when prices are changed according to the law of supply and demand (Kaldor, 1934; Arrow and Hahn, 1971, pp. 264, 282). Historically speaking, however, this usage is somewhat confusing, since recontracting is originally due to Edgeworth who developed it in a direction different from that in which Walras developed his tâtonnement (Walker, 1973).

In 1935 the Japanese economist Shibata criticized Bohm-Bawerk's discussion to deny the role of power on market and argued that the validity of Bohm-Bawerk's conclusion depends crucially on his special assumptions on the structure of production in his theory of interest. Unfortunately, both Bohm-Bawerk and Shibata used in their discussions an untenable supposition that the value of heterogeneous capital, which depends on values of endogenous variables, can be given exogenously. Our aim is to reconsider the problem in the light of a new interpretation of Bohm-Bawerk's theory of interest, which is free from this supposition.

Studies of von Thunen in Japan have a long history and suggest an interpretation of von Thunen's theory, which is different from those of recent studies of von Thunen in the English speaking world. A simple model of a stationary economy is constructed on the basis of such an interpretation, in which workers are assumed to be free to save in their life-cycles though the level of consumption of working families is kept at the subsistence level. Von Thunen's famous formula of the natural wage is derived from the model, which is free from the criticisms given to von Thunen's original model.

to estimate the reaction function of the rival which will be shifted as it learns from experiences, i. e., observations of facts. In other words, an equilibrium is possible where conjectures are justified by results with the coefficient of conjectural variation explained in terms of parameters in demand and cost functions, if each firm is assumed to behave as the leader in the sense of Stackelberg. The stability of Stackelberg equilibrium is shown when the marginal costs increase sharply relative to the slope of the demand curve.

Samuelson’s work on stability (collected in Samuelson 1947, pp. 257–349) contains the first rigorous attempt to analyze the dynamic stability of Walrasian tâtonnement. It is interesting, however, to see that a recent trend in microeconomics called non-Walrasian economics has developed, via the stability analysis of the so-called non-tâtonnement process, from this study of the most typically Walrasian problem.

This chapter discusses the significance of Gossen's work. Gossen anticipated not only the theory of marginal utility of which Jevons shared the fame with Walras and Menger, but also Jevons's theory of exchange in the market, which was based on a view of the market quite different from that of Cournot and Walras, and was succeeded and developed by Edgeworth. Gossen's theory of exchange can be considered as the predecessor of the theory of Jevons, and also of Edgeworth. Gossen considered the case of a two-individual two-good exchange, with highly special assumptions on the shape of the marginal utility curves and on the initially available quantities of goods to each individual. It clearly shows, however, a view of markets which is different from that of Cournot and Walras and is similar to those of Jevons and Edgeworth.

In his Mathematical Psychics (1881), Edgeworth demonstrated in so-called Edgeworth equivalence theorem that the outcome of an exchange economy where traders act cooperatively is identical to the Walrasian equilibrium of perfect competition where traders act non-cooperatively as price-takers, when the number of traders of the same type is infinitely large. Certainly it clarifies an implication of Walrasian economics and is a pioneering work of the recent theory of a large economy composed of infinitely many small traders. 1When the number of traders is limited, Edgeworth argued, traders themselves cannot determine the outcome of an exchange economy cooperatively and the utilitarian principle should be introduced to arbitrate disagreement between traders. Since the infinity of identical traders is a sufficient condition in the proof of Edgeworth’s equivalence theorem, however, a natural question is whether it is also a necessary one. If it is not necesary, what is essential for Walrasian economies is not so much the infinity of the number of small traders as perfect information and no friction, also assumed in Edgeworth’s theorem.

Among the many contributions to economic science made by Arrow, one of the most important is certainly the proof of the existence of an equilibrium for a competitive economy.1 The case where no equilibrium exists even though indifference curves, production functions, and so on, are fairly well behaved is a useful one to show the necessity of proving the existence of equilibrium. Mill (1869) indicates that one of the first examples of the non-existence of equilibrium consists of the counter examples to equilibrium theory given in W. T. Thornton’s On Labour (1869) though Thornton himself was concerned not so much with the non-existence of equilibrium as the possibility of trade at disequilibrium prices. The Thornton-Mill examples of the non-existence of equilibrium are remarkable because they are due to the discontinuity of demand curves; other unsuccessful attempts to show the non-existence of equilibrium failed because of their assumption of continuity. The Thornton-Mill examples, as well as the example of Wald (1951), are, however, not so serious to equilibrium theory, if we consider, not the Walrasian tâtonnement with recontract, but the non-tâtonnement without recontract. From such non-tâtonnement point of view, a truly important example of the non-existence of equilibrium is the one given by Arrow, that is, the case where a Pareto optimal allocation cannot be viewed as a competitive equilibrium.2

This chapter discusses Alfred Marshall's Economics. His theory could be said to be dominantly influential to the development of economics in England after the marginal revolution. Unlike Jevons, who openly attacked classical economics, Marshall did not deny the significance of classical economics, but tried to make it more generalized. J.R. Hicks tried to combine Walrasian and Marshallian traditions in his Value and Capital . Hicks insists that Walras and Marshall differ in interest, the former in principles and the latter in practical applications. Even if one is interested in principles only, however, Marshall's contributions are necessary complements to Walrasian ones. Firstly, the time structure of Marshallian equilibria—market day, short-run, and long-run—clarifies the economic significance of time elements, and gives useful suggestions on how to generalize the static and timeless Walrasian general equilibrium theory dynamically. The chapter also presents a problem related to a Marshallian dynamic equilibrium of the industry which is based on the biological analogy.

(a) In the methodological discussion, Newman states that global stability may be too stringent a condition to impose on the system, and that it is not always to be preferred to local stability as a condition. It seems to me that Newman implicitly assumes the uniqueness of equilibrium. If the equilibrium is unique, global stability implies local stability; and, since in every period the market-clearing process is considered to start from historically given prices close to the equilibrium prices rather than from " prix cries par hasard ",2 local stability is what is needed for any dynamic model intended to represent reality. As Arrow justly points out, however, the theory of the uniqueness of equilibrium is fragmentary and poorly developed.3 Even if the equilibrium price vector established by tatonnement is unique (as in the case of gross substitutes), it cannot be unique when established by the general process in which (contrary to tatonnement) changes in the distribution of commodity stocks among individuals take place through exchange transactions.4 In studies of global stability we are concerned with the stability of a system as a whole (say a competitive economy) rather than with the stability of the equilibrium. If the uniqueness of equilibrium is not assumed, it is possible for there to be multiple equilibria, none of which is completely stable from the local point of view (i.e., attracts all neighbouring points), while the system in its entirety is globally stable (i.e., approaches some equilibrium). In this case global stability is a less stringent condition than local stability, and is more appropriately to be studied as a property of a realistic dynamic model of the economy.

and to estimate their magnitude, by giving a rational basis to extrapolative expectations. Rationality implies the use of economic theory, considering the cost of information and computation. Extrapolative expectations are derived as the prediction of the equilibrium by the use of estimated excess demand functions, and it is shown that the coefficients of expectations thus derived are such that the system of multiple markets is stable when gross substitutability and tatonnement are also assumed. THE DYNAMIC stability of multiple markets was studied in [3] under the assumption of extrapolative expectations, with the result that stability depends, when gross substitutability and tatonnement are assumed, on the magnitude of the coefficients of expectations whose economic meaning is not necessarily clear. In this note, we shall give some rational basis to extrapolative expectations. The rational expectation hypothesis advanced by Muth [4] is that expectations are essentially the same as the predictions of the relevant economic theory; that the economy generally does not waste information; and that expectations depend specifically on the structure of the entire system. However, since there is cost of information and computation, expectations may also be called rational when they are formed as the prediction based on a simplified and approximated version of the economic theory, using only limited amounts of information on a part of the system. Extrapolative expectations will be derived below as the prediction of the equilibrium by the use of estimated excess demand functions, and it will be shown that the coefficients of expectations thus derived are such that the system of multiple markets is stable when gross substitutability and tatonnement are assumed.

THE THEORY of the general competitive equilibrium, as developed by Leon Walras [64], has recently been reformulated in terms of fairly advanced mathematical methods. The first problem studied extensively was concerned with the conditions under which a competitive equilibrium exists for a model of general equilibrium. Among contributions to the existence problem, Wald [63], Arrow and Debreu [5], McKenzie [35], Nikaido [48], Gale [18], and Negishi [44] may be cited. The optimality problem of a competitive equilibrium was also investigated by, e.g., Arrow [2], Debreu [15, 16], and, in

THE STABILITY of dynamic systems has been extensively studied by Arrow and Nerlove [3] and Arrow and Hurwicz [2] under the assumption of gross substitutability and adaptive expectation. In their systems, it is assumed that a current price is changed by current excess demand, and that excess demand is a function of all current and expected future prices, the latter being formed adaptively to the past history of prices. One should note that they explicitly or implicitly assumed that excess demands were independent of transactions among individuals. Two alternative interpretations of this assumption were suggested by Arrow and Hurwicz [2]. One is the recontract interpretation of the Walrasian ta'tonnement, in which no transactions actually take place during the process until equilibrium is achieved. The second interpretation is that transactions can indeed take place, but that they do not affect decisions taken at any subsequent period, because all the commodities are assumed to be instantaneously perishable, and the utility for present and future goods is also assumed to be the sum of utilities of present goods and of future goods, the utility functions being of the same mathematical form. The second interpretation seems more interesting to us, since, once the assumption of perishability is made, the dynamic system can describe the changes of actual prices over time.2 In the following, however, we are concerned with another polar case, dropping the assumption of perishability and substituting another extreme assumption, namely, that all the commodities are permanently durable with negligible depreciation by consumption. We shall keep the hypothesis that the utility function for present goods and that for future goods are of mathematically same form and are independent of the argu